Informative Path Planning for Active Regression With Gaussian Processes via Sparse Optimization

摘要

We study informative path planning for active regression in Gaussian Processes (GP). Here, a resource constrained robot team collects measurements of an unknown function, assumed to be a sample from a GP, with the goal of minimizing the trace of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula>-weighted expected squared estimation error covariance (where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> is a positive semidefinite matrix) resulting from the GP posterior mean. While greedy heuristics are a popular solution in the case of length constrained paths, it remains a challenge to compute <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">optimal</i> solutions in the discrete setting subject to routing constraints. We show that this challenge is surprisingly easy to circumvent. Using the optimality of the posterior mean for a class of functions of the squared loss yields an exact formulation as a mixed integer program. We demonstrate that this approach finds optimal solutions in a variety of settings in seconds and when terminated early, it finds sub-optimal solutions of higher quality than existing heuristics.